Kepler's first law
First consider the elliptical orbits described in Kepler’s first law. Figure 13.18 shows the geometry of an ellipse. The longest dimension is the major axis, with half-length a; this half-length is called the semi-major axis. The sum of the dis-tances from S to P and from S to P is the same for all points on the curve. S and S are the foci (plural of focus). The sun is at S (not at the center of the ellipse) and the planet is at P; we think of both as points because the size of each is very small in comparison to the distance between them. There is nothing at the other focus, S.
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The distance of each focus from the center of the ellipse is ea, where e is a dimensionless number between 0 and 1 called the eccentricity. If e=0, the two foci coincide and the ellipse is a circle. The actual orbits of the planets are fairly circular; their eccentricities range from 0.007 for Venus to 0.206 for Mercury. (The earth’s orbit has e=0.017.) The point in the planet’s orbit closest to the sun is the perihelion, and the point most distant is the aphelion.
Newton showed that for a body acted on by an attractive force proportional to 1/r2, the only possible closed orbits are a circle or an ellipse; he also showed that open orbits (trajectories 6 and 7 in Fig. 13.14) must be parabolas or hyperbolas. These results can be derived from Newton’s laws and the law of gravitation, together with a lot more differential equations than we’re ready for.